class: center, middle, inverse, title-slide .title[ # ISA 444: Business Forecasting ] .subtitle[ ## 05: Time Series Summaries ] .author[ ### <br>Fadel M. Megahed, PhD <br><br> Endres Associate Professor <br> Farmer School of Business<br> Miami University<br><br> <a href="https://twitter.com/FadelMegahed"><svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"></path></svg> <span class="citation">@FadelMegahed</span></a> <br> <a href="https://github.com/fmegahed/"><svg viewBox="0 0 496 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"></path></svg> fmegahed</a> <br> <a href="mailto:fmegahed@miamioh.edu"><svg viewBox="0 0 512 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"></path></svg> fmegahed@miamioh.edu</a><br> <a href="https://calendly.com/fmegahed"><svg viewBox="0 0 384 512" style="height:1em;position:relative;display:inline-block;top:.1em;fill:white;" xmlns="http://www.w3.org/2000/svg"> <path d="M202.021 0C122.202 0 70.503 32.703 29.914 91.026c-7.363 10.58-5.093 25.086 5.178 32.874l43.138 32.709c10.373 7.865 25.132 6.026 33.253-4.148 25.049-31.381 43.63-49.449 82.757-49.449 30.764 0 68.816 19.799 68.816 49.631 0 22.552-18.617 34.134-48.993 51.164-35.423 19.86-82.299 44.576-82.299 106.405V320c0 13.255 10.745 24 24 24h72.471c13.255 0 24-10.745 24-24v-5.773c0-42.86 125.268-44.645 125.268-160.627C377.504 66.256 286.902 0 202.021 0zM192 373.459c-38.196 0-69.271 31.075-69.271 69.271 0 38.195 31.075 69.27 69.271 69.27s69.271-31.075 69.271-69.271-31.075-69.27-69.271-69.27z"></path></svg> Automated Scheduler for Office Hours</a><br><br> ] .date[ ### Spring 2023 ] --- # Quick Refresher from Last Class ✅ Examine a line chart for trends, seasonality, and cycles. ✅ Explain the grammar of graphics and how it can be used to create time series plots in <svg aria-hidden="true" role="img" viewBox="0 0 581 512" style="height:1em;width:1.13em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:#C3142D;overflow:visible;position:relative;"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg>. ✅ Create interactive time-series plots by using the [plotly](https://plotly.com/ggplot2/getting-started/) package <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:gold;overflow:visible;position:relative;"><path d="M50.7 58.5L0 160H208V32H93.7C75.5 32 58.9 42.3 50.7 58.5zM240 160H448L397.3 58.5C389.1 42.3 372.5 32 354.3 32H240V160zm208 32H0V416c0 35.3 28.7 64 64 64H384c35.3 0 64-28.7 64-64V192z"/></svg>. --- # Learning Objectives for Today's Class - Use numerical summaries to describe a time series. - Explain what do we mean by correlation. - Apply transformations to a time series. --- class: inverse, center, middle # A Summarizing Time-Series Data --- # Measures of Average **Mean:** Given a set of `\(n\)` values `\(Y_1, \, Y_2, \, \dots, \, Y_n\)`, the arithmetic mean can be computed as: `$$\bar{Y} = \frac{Y_1 + Y_2 + \dots + Y_n}{n} = \frac{1}{n}\sum_{i=1}^{i=n}Y_i.$$` <br> **Order Statistics:** Given a set of `\(n\)` values `\(Y_1, \, Y_2, \, \dots, \, Y_n\)`, we place them in an ascending order to define the order statistics, written as `\(Y_{(1)}, \, Y_{(2)}, \, \dots, \, Y_{(n)}.\)` **Median:** - If `\(n\)` is odd, `\(n = 2m + 1\)` and the median is `\(Y_{(m+1)}\)`. - If `\(n\)` is even, `\(n = 2m\)` and the median is the average of the two middle numbers, i.e., `\(\frac{1}{2}[Y_{(m)} + Y_{(m+1)}]\)`. --- # Measures of Variation The **range** denotes the difference between the largest and smallest value in a sample: `$$Range = Y_{(n)} - Y_{(1)}.$$` The **deviation** is defined as the difference between a given observation `\(Y_i\)` and the mean `\(\bar{Y}\)`. The **mean absolute deviation (MAD)** is the average deviations about the mean, irrespective of their sign: $$ \text{MAD} = \frac{\sum_{i=1}^{i=n}|d_i|}{n}. $$ The **variance** is the average of the squared deviations around the mean: $$ S^2 = \frac{\sum_{i=1}^{i=n}d_i^2}{n-1}. $$ --- # The GameStop Short Squeeze .center[ <div class="plotly html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-e5ad35a6e9dddfaa5ff7" style="width:600px;height:300px;"></div> <script type="application/json" 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(y) ","x"],"6d606ce06eb":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> ] --- ## Summarizing the GME Short Squeeze: Avg/Var Measures .pull-left-2[ .font80[ ```r gme_get = tidyquant::tq_get(x = 'GME', from = '2020-01-01') |> dplyr::select(date, adjusted) |> dplyr::mutate( year = lubridate::year(date), month = lubridate::month(date, label = T) ) gme_get ``` ] ] .pull-right-2[ ``` ## # A tibble: 779 × 4 ## date adjusted year month ## <date> <dbl> <dbl> <ord> ## 1 2020-01-02 1.58 2020 Jan ## 2 2020-01-03 1.47 2020 Jan ## 3 2020-01-06 1.46 2020 Jan ## 4 2020-01-07 1.38 2020 Jan ## 5 2020-01-08 1.43 2020 Jan ## 6 2020-01-09 1.39 2020 Jan ## 7 2020-01-10 1.36 2020 Jan ## 8 2020-01-13 1.36 2020 Jan ## 9 2020-01-14 1.18 2020 Jan ## 10 2020-01-15 1.15 2020 Jan ## # … with 769 more rows ``` ] --- count: false ## Summarizing the GME Short Squeeze: Avg/Var Measures .pull-left-2[ .font80[ ```r gme_get = tidyquant::tq_get(x = 'GME', from = '2020-01-01') |> dplyr::select(date, symbol, adjusted) |> dplyr::mutate( year = lubridate::year(date), month = lubridate::month(date, label = T) ) *gme_summary = * gme_get |> * dplyr::group_by(symbol) gme_summary ``` ] ] .pull-right-2[ ``` ## # A tibble: 779 × 5 ## # Groups: symbol [1] ## date symbol adjusted year month ## <date> <chr> <dbl> <dbl> <ord> ## 1 2020-01-02 GME 1.58 2020 Jan ## 2 2020-01-03 GME 1.47 2020 Jan ## 3 2020-01-06 GME 1.46 2020 Jan ## 4 2020-01-07 GME 1.38 2020 Jan ## 5 2020-01-08 GME 1.43 2020 Jan ## 6 2020-01-09 GME 1.39 2020 Jan ## 7 2020-01-10 GME 1.36 2020 Jan ## 8 2020-01-13 GME 1.36 2020 Jan ## 9 2020-01-14 GME 1.18 2020 Jan ## 10 2020-01-15 GME 1.15 2020 Jan ## # … with 769 more rows ``` ] --- count: false ## Summarizing the GME Short Squeeze: Avg/Var Measures .pull-left-2[ .font80[ ```r gme_get = tidyquant::tq_get(x = 'GME', from = '2020-01-01') |> dplyr::select(date, symbol, adjusted) |> dplyr::mutate( year = lubridate::year(date), month = lubridate::month(date, label = T) ) gme_summary = gme_get |> dplyr::group_by(symbol) |> * dplyr::summarise( * ajusted_avg = mean(adjusted), * adjusted_med = median(adjusted), * adjusted_var = var(adjusted), * adjusted_sd = sd(adjusted) * ) gme_summary |> t() # transposing for printout ``` ] ] .pull-right-2[ ``` ## [,1] ## symbol "GME" ## ajusted_avg "24.44359" ## adjusted_med "26.05" ## adjusted_var "358.923" ## adjusted_sd "18.94526" ``` ] --- count: false ## Summarizing the GME Short Squeeze: Avg/Var Measures .pull-left-2[ .font80[ ```r gme_get = tidyquant::tq_get(x = 'GME', from = '2020-01-01') |> dplyr::select(date, symbol, adjusted) |> dplyr::mutate( year = lubridate::year(date), month = lubridate::month(date, label = T) ) gme_summary = gme_get |> * dplyr::group_by(symbol, year) |> dplyr::summarise( ajusted_avg = mean(adjusted), adjusted_med = median(adjusted), adjusted_var = var(adjusted), adjusted_sd = sd(adjusted) ) gme_summary ``` ] ] .pull-right-2[ ``` ## # A tibble: 4 × 6 ## # Groups: symbol [1] ## symbol year ajusted_avg adjusted_med adjusted_var adjusted_sd ## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> ## 1 GME 2020 1.79 1.19 1.17 1.08 ## 2 GME 2021 42.4 44.5 207. 14.4 ## 3 GME 2022 29.6 28.9 32.9 5.74 ## 4 GME 2023 19.9 20.5 4.54 2.13 ``` ] --- count: false ## Summarizing the GME Short Squeeze: Avg/Var Measures .pull-left-2[ .font80[ ```r gme_get = tidyquant::tq_get(x = 'GME', from = '2020-01-01') |> dplyr::select(date, symbol, adjusted) |> dplyr::mutate( year = lubridate::year(date), month = lubridate::month(date, label = T) ) gme_summary = gme_get |> * dplyr::group_by(symbol, year, month) |> dplyr::summarise( ajusted_avg = mean(adjusted), adjusted_med = median(adjusted), adjusted_var = var(adjusted), adjusted_sd = sd(adjusted) ) print(gme_summary, n=15) ``` ] ] .pull-right-2[ ``` ## # A tibble: 38 × 7 ## # Groups: symbol, year [4] ## symbol year month ajusted_avg adjusted_med adjusted_var adjusted_sd ## <chr> <dbl> <ord> <dbl> <dbl> <dbl> <dbl> ## 1 GME 2020 Jan 1.22 1.16 0.0321 0.179 ## 2 GME 2020 Feb 0.981 1.00 0.00410 0.0640 ## 3 GME 2020 Mar 1.00 0.992 0.00522 0.0722 ## 4 GME 2020 Apr 1.15 1.20 0.0742 0.272 ## 5 GME 2020 May 1.16 1.12 0.0164 0.128 ## 6 GME 2020 Jun 1.15 1.14 0.00545 0.0738 ## 7 GME 2020 Jul 1.03 1.03 0.00128 0.0358 ## 8 GME 2020 Aug 1.20 1.16 0.0188 0.137 ## 9 GME 2020 Sep 2.13 2.17 0.124 0.352 ## 10 GME 2020 Oct 3.04 3.03 0.221 0.470 ## 11 GME 2020 Nov 3.11 2.92 0.181 0.425 ## 12 GME 2020 Dec 4.15 4.06 0.410 0.640 ## 13 GME 2021 Jan 19.9 9.78 648. 25.5 ## 14 GME 2021 Feb 17.9 13.1 117. 10.8 ## 15 GME 2021 Mar 47.1 48.6 133. 11.5 ## # … with 23 more rows ``` ] --- class: inverse, center, middle # Correlation --- # The Pearson Correlation Coefficient - **Correlation:** measures the strength of the **linear relationship** between two quantitative variables. - It can be computed using the `cor()` from base R. Mathematically speaking, the pearson correlation coefficient, `\(r\)`, can be computed as `$$r = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum_{i=1}^{n}(X_i - \bar{X})^2 \sum_{i=1}^{n}(Y_i - \bar{Y})^2}}$$` - Do **not** use the Pearson Correlation coefficient if both variables are not quantitative. Instead, refer to the `mixed.cor()` from the [psch package](https://personality-project.org/r/psych/help/mixed.cor.html) to compute the correlations for mixtures of continuous, polytomous, and/or dichotomous variables. - You should supplement **any descriptive summaries with visualizations** to ensure that you are able to interpret the computations correctly. --- ## Supplement Summaries with Viz: Anscombe's Dataset **In a seminal paper, Anscombe stated:** > **Few of us escape being indoctrinated with these notions:** > - numerical **calculations are exact, but graphs are rough**; > - for any particular kind of **statistical data there is just one set of calculations constituting a correct statistical analysis**; > - performing **intricate calculations is virtuous**, whereas **actually looking at the data is cheating**. He proceeded by stating that > a computer should **make both calculations and graphs**. Both sorts of output should be studied; each will contribute to understanding. Now, let us consider his four datasets, each consisting of eleven (x,y) pairs. .footnote[ <html> <hr> </html> **Source:** Anscombe, Francis J. 1973. "Graphs in Statistical Analysis." *The American Statistician* 27 (1): 17–21. ([PDF Link](https://www.sjsu.edu/faculty/gerstman/StatPrimer/anscombe1973.pdf)). --- count: false ## Supplement Summaries with Viz: Anscombe's Dataset .font80[ <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-87721d10ea330e46143b" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-87721d10ea330e46143b">{"x":{"filter":"none","vertical":false,"data":[[10,8,13,9,11,14,6,4,12,7,5],[10,8,13,9,11,14,6,4,12,7,5],[10,8,13,9,11,14,6,4,12,7,5],[8,8,8,8,8,8,8,19,8,8,8],[8.04,6.95,7.58,8.81,8.33,9.96,7.24,4.26,10.84,4.82,5.68],[9.14,8.14,8.74,8.77,9.26,8.1,6.13,3.1,9.13,7.26,4.74],[7.46,6.77,12.74,7.11,7.81,8.84,6.08,5.39,8.15,6.42,5.73],[6.58,5.76,7.71,8.84,8.47,7.04,5.25,12.5,5.56,7.91,6.89]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>x1<\/th>\n <th>x2<\/th>\n <th>x3<\/th>\n <th>x4<\/th>\n <th>y1<\/th>\n <th>y2<\/th>\n <th>y3<\/th>\n <th>y4<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":11,"dom":"tip","columnDefs":[{"className":"dt-right","targets":[0,1,2,3,4,5,6,7]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[10,11,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":[],"jsHooks":[]}</script> ] --- count: false ## Supplement Summaries with Viz: Anscombe's Dataset .font80[ <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-97daa3a6979e4dc95254" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-97daa3a6979e4dc95254">{"x":{"filter":"none","vertical":false,"data":[["I","II","III","IV"],[9,9,9,9],[3.32,3.32,3.32,3.32],[7.5,7.5,7.5,7.5],[2.03,2.03,2.03,2.03],[0.82,0.82,0.82,0.82]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>set<\/th>\n <th>x.mean<\/th>\n <th>x.sd<\/th>\n <th>y.mean<\/th>\n <th>y.sd<\/th>\n <th>corr<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"dom":"tip","columnDefs":[{"className":"dt-right","targets":[1,2,3,4,5]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":[],"jsHooks":[]}</script> ] --- count: false ## Supplement Summaries with Viz: Anscombe's Dataset <img src="data:image/png;base64,#05_ts_summary_files/figure-html/anscombe4-1.png" width="70%" style="display: block; margin: auto;" /> --- # Kahoot Competition #02 To assess your understanding and retention of the topics covered so far, you will **compete in a Kahoot competition (consisting of 6 questions)**: - Go to <https://kahoot.it/> - Enter the game pin, which will be shown during class - Provide your first (preferred) and last name - Answer each question within the allocated time window (**fast and correct answers provide more points**) **Winning the competition involves having as many correct answers as possible AND taking the shortest duration to answer these questions.** The winner <svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:gold;overflow:visible;position:relative;"><path d="M400 0H176c-26.5 0-48.1 21.8-47.1 48.2c.2 5.3 .4 10.6 .7 15.8H24C10.7 64 0 74.7 0 88c0 92.6 33.5 157 78.5 200.7c44.3 43.1 98.3 64.8 138.1 75.8c23.4 6.5 39.4 26 39.4 45.6c0 20.9-17 37.9-37.9 37.9H192c-17.7 0-32 14.3-32 32s14.3 32 32 32H384c17.7 0 32-14.3 32-32s-14.3-32-32-32H357.9C337 448 320 431 320 410.1c0-19.6 15.9-39.2 39.4-45.6c39.9-11 93.9-32.7 138.2-75.8C542.5 245 576 180.6 576 88c0-13.3-10.7-24-24-24H446.4c.3-5.2 .5-10.4 .7-15.8C448.1 21.8 426.5 0 400 0zM48.9 112h84.4c9.1 90.1 29.2 150.3 51.9 190.6c-24.9-11-50.8-26.5-73.2-48.3c-32-31.1-58-76-63-142.3zM464.1 254.3c-22.4 21.8-48.3 37.3-73.2 48.3c22.7-40.3 42.8-100.5 51.9-190.6h84.4c-5.1 66.3-31.1 111.2-63 142.3z"/></svg> of the competition will receive a $10 Starbucks gift card. Good luck!!! .footnote[ <html> <hr> </html> **P.S:** The Kahoot competition will have **no impact on your grade**. It is a **fun** way of assessing your knowledge, motivating you to ask questions about topics covered that you do not have a full understanding of it, and providing me with some data to pace class. ] --- class: inverse, center, middle # Transformations --- # Guidelines for Transforming Time Series Data <div class="figure" style="text-align: center"> <img src="data:image/png;base64,#../../figures/transformations.png" alt="A classification of common transformation approaches for time series data" width="100%" /> <p class="caption">A classification of common transformation approaches for time series data</p> </div> .footnote[ <html> <hr> </html> My (incomplete) attempt to provide you with a taxonomy for time series data transformations. ] --- # Stablize the Mean: Differencing The plot below shows the number of murdered women per 100,000 people in the U.S. From the plot, we can see that the ts is not stationary. <img src="data:image/png;base64,#05_ts_summary_files/figure-html/murdered_women-1.png" width="70%" style="display: block; margin: auto;" /> --- count:false # Stablize the Mean: Differencing The plot below shows the **first nonseasonal difference**. From the plot, we can see that differencing has reduced the nonstationary nature of the time-series. <img src="data:image/png;base64,#05_ts_summary_files/figure-html/murdered_women2-1.png" width="70%" style="display: block; margin: auto;" /> --- # Computing the First Nonseasonal Difference The change in the time series from one period to the next is known as the first nonseasonal difference. It can be computed as follows: `$$DY_t = Y_t - Y_{t-1}$$` .pull-left-2[ .font80[ ```r women_murdered_filtered = women_murdered |> # dataset read in previous lines of code dplyr::filter(year > 2000) print(women_murdered_filtered) ``` ] ] .pull-right-2[ ``` ## # A tibble: 6 × 3 ## country year murders_per_100000 ## <chr> <dbl> <dbl> ## 1 United States 2001 2.77 ## 2 United States 2002 2.64 ## 3 United States 2003 2.57 ## 4 United States 2004 2.53 ## 5 United States 2005 2.6 ## 6 United States 2006 2.58 ``` ] --- count:false # Computing the First Nonseasonal Difference The change in the time series from one period to the next is known as the first nonseasonal difference. It can be computed as follows: `$$DY_t = Y_t - Y_{t-1}$$` .pull-left-2[ .font80[ ```r women_murdered_filtered = women_murdered |> # dataset read in previous lines of code dplyr::filter(year > 2000) *women_murdered_filtered = * women_murdered_filtered |> * dplyr::mutate( * diff1 = murders_per_100000 - dplyr::lag(murders_per_100000, n = 1), * diff2 = c(NA, diff(murders_per_100000, lag = 1)) * ) print(women_murdered_filtered) ``` ] ] .pull-right-2[ ``` ## # A tibble: 6 × 5 ## country year murders_per_100000 diff1 diff2 ## <chr> <dbl> <dbl> <dbl> <dbl> ## 1 United States 2001 2.77 NA NA ## 2 United States 2002 2.64 -0.130 -0.130 ## 3 United States 2003 2.57 -0.0700 -0.0700 ## 4 United States 2004 2.53 -0.0400 -0.0400 ## 5 United States 2005 2.6 0.0700 0.0700 ## 6 United States 2006 2.58 -0.0200 -0.0200 ``` ] --- # Computing the First Seasonal Difference If your data exhibits a seasonal pattern, as illustrated in Slides 6 and 18 in [03_ts_viz.html](https://fmegahed.github.io/isa444/spring2023/class03/03_ts_viz.html), you should employ a **seasonal differencing approach**, you should subtract the difference between an observation and the previous observation from the same season. Let `\(m\)` denote the number of seasons, e.g. `\(m=4\)` for quarterly data. In such a case, the seasonal difference is computed as follows: `$$DY_{t-m} = Y_t - Y_{t-m}$$` **Note:** In <svg aria-hidden="true" role="img" viewBox="0 0 581 512" style="height:1em;width:1.13em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:#C3142D;overflow:visible;position:relative;"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg>, this can be computed by assigning the `\(x\)` argument in the `dplyr::lag()` to `\(m\)`, or by setting the `\(lag\)` argument in the `diff()` to `\(m\)`. --- # Stablize the Variance: Power Transformations .left-code[ .font80[ ```r # The built-in JohnsonJohnson dataset forecast::autoplot(JohnsonJohnson) + ggplot2::geom_point() + # adding points ggplot2::scale_x_continuous(breaks = scales::pretty_breaks(n=6)) + ggplot2::scale_y_continuous(breaks = scales::pretty_breaks(n = 6)) + ggplot2::theme_bw() ``` ] ] .right-plot[ <img src="data:image/png;base64,#05_ts_summary_files/figure-html/jj1_out-1.png" width="70%" style="display: block; margin: auto;" /> ] --- count: false # Stablize the Variance: Power Transformations .left-code[ .font80[ ```r # The built-in JohnsonJohnson dataset *forecast::autoplot(sqrt(JohnsonJohnson)) + ggplot2::geom_point() + # adding points ggplot2::scale_x_continuous(breaks = scales::pretty_breaks(n=6)) + ggplot2::scale_y_continuous(breaks = scales::pretty_breaks(n = 6)) + ggplot2::theme_bw() ``` ] ] .right-plot[ <img src="data:image/png;base64,#05_ts_summary_files/figure-html/jj2_out-1.png" width="70%" style="display: block; margin: auto;" /> ] --- count: false # Stablize the Variance: Power Transformations .left-code[ .font80[ ```r # The built-in JohnsonJohnson dataset *forecast::autoplot(log(JohnsonJohnson)) + ggplot2::geom_point() + # adding points ggplot2::scale_x_continuous(breaks = scales::pretty_breaks(n=6)) + ggplot2::scale_y_continuous(breaks = scales::pretty_breaks(n = 6)) + ggplot2::theme_bw() ``` ] ] .right-plot[ <img src="data:image/png;base64,#05_ts_summary_files/figure-html/jj3_out-1.png" width="70%" style="display: block; margin: auto;" /> ] --- # A Note on the Log Transform The log transformation can be computed as follows: `$$L_t = \ln{(Y_t)}$$` Note that the `log()` in R takes the natural logarithm as its default base, i.e., would transform a variable/statistic based on the above equation. The reverse transformation using the exponential function is: `$$e^{L_t} = e^{\ln{(Y_t})} = Y_t$$` --- count: false # The Log Transform - The primary purpose of the log transform is to **convert exponential growth into linear growth.** - The transform often has the **secondary purpose of balancing the variance.** - Difference in logs and growth rate transformations produce similar results and interpretations (see next slides). --- # Stabilizing the Mean and Variance The **first nonseasonal difference in logarithms** represents the logarithm of the ratio `$$L_t = \ln{(\frac{Y_t}{Y_{t-1}})} = \ln{(Y_t)} - \ln{(Y_{t-1})}$$` In the absence of seasonality, the **growth rate** for a time series is given by `$$GY_t = 100 \frac{Y_t - Y_{t-1}}{Y_{t-1}}$$` --- count: false # Stabilizing the Mean and Variance .left-code[ .font80[ ```r # The built-in JohnsonJohnson dataset forecast::autoplot( * log(JohnsonJohnson) - log(stats::lag(JohnsonJohnson)) ) + ggplot2::geom_point() + # adding points ggplot2::scale_x_continuous(breaks = scales::pretty_breaks(n=6)) + ggplot2::scale_y_continuous(breaks = scales::pretty_breaks(n = 6)) + ggplot2::theme_bw() ``` ] ] .right-plot[ <img src="data:image/png;base64,#05_ts_summary_files/figure-html/jj4_out-1.png" width="70%" style="display: block; margin: auto;" /> ] --- count: false # Stabilizing the Mean and Variance .left-code[ .font80[ ```r # The built-in JohnsonJohnson dataset forecast::autoplot( * (JohnsonJohnson - stats::lag(JohnsonJohnson))/ stats::lag(JohnsonJohnson) ) + ggplot2::geom_point() + # adding points ggplot2::scale_x_continuous(breaks = scales::pretty_breaks(n=6)) + ggplot2::scale_y_continuous(breaks = scales::pretty_breaks(n = 6)) + ggplot2::theme_bw() ``` ] ] .right-plot[ <img src="data:image/png;base64,#05_ts_summary_files/figure-html/jj5_out-1.png" width="70%" style="display: block; margin: auto;" /> ] --- # A Practical Note about Growth Rates <div class="countdown" id="timer_b4c92a24" data-update-every="1" tabindex="0" style="top:0;right:0;font-size:2em;"> <div class="countdown-controls"><button class="countdown-bump-down">−</button><button class="countdown-bump-up">+</button></div> <code class="countdown-time"><span class="countdown-digits minutes">05</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code> </div> .panelset[ .panel[.panel-name[Activity] > Over the next 5 minutes, please answer the question in each tab. ] .panel[.panel-name[Q1] - **Question 1:** Let us say that an investor purchased 10 stocks of \$GME, on 2021-01-29, at 325/stock. The next trading day, 2021-02-01, the GME stock closed at $225. Compute the growth rate in their portfolio worth (assuming it only has the GME stock) over this time period. .can-edit.key-activity4_q1[ **What is their growth rate?** .font70[(Insert below)] - Edit me ] ] .panel[.panel-name[Q2] - **Question 2:** Let us say that the growth rate, `\(GY_t = -g\)`. Now let us assume that the GME stock went up by `\(g\)` (i.e., if it went down 10%, it increased by 10% over the next trading day). What is the value of the investor's portfolio by stock market closing on 2021-02-02? .can-edit.key-activity4_q2[ **What is their growth rate?** .font70[(Insert below)] - Edit me ] ] ] --- # A Live Demo In this live coding session, we will capitalize on the `mutate()` from [tidyverse](https://www.tidyverse.org/) to create transformations for multiple time series. Specifically, we will use the `tq_get()` from [tidyquant](https://business-science.github.io/tidyquant/) to extract data about the following cryptocurrencies (a) [Cardano](https://cardano.org/) (ADA), (b) [Chainlink](https://chain.link/) (LINK), and (c) [Zilliqa](https://www.zilliqa.com/) (ZIL). We will compute: - Growth Rates - Natural log - Log Differences - `\([0-1]\)` Scaling Obviously, we will have to ensure that these transformations are computed for each coin separately. For the purpose of this activity, let us extract the data from 2023-01-01 to 2023-02-05. --- class: inverse, center, middle # Recap --- # Summary of Main Points By now, you should be able to do the following: - Use numerical summaries to describe a time series. - Explain what do we mean by correlation. - Apply transformations to a time series. --- # Things to Do to Prepare for Our Next Class - Go over your notes and read through [Chapter 2.1-2.5 of our reference book](https://cdn.shopify.com/s/files/1/0859/4364/files/Part_I_POBF-_A_First_Course_in_Forecasting_1.pdf?612). - Complete [Assignment 04](https://miamioh.instructure.com/courses/188655/assignments/2377018) on Canvas.